Solutions to a Pillai-type equation involving Tribonacci numbers and S-units

Let $\{T_n{\}}_{n\ge 0}$ be the sequence of Tribonacci numbers. In this paper, we study the exponential Diophantine equation $T_n-2^x{3}^y=c$, for $n,x,y\in {\mathbb{Z}}_{\ge 0}$. In particular, we show that there is no integer $c$ with at least six representations of the form $T_n-2^x{3}^y$.